Numerical computation of nonlinear shock wave … · Numerical computation of nonlinear shock wave...
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Ain Shams Engineering Journal (2015) 6, 605–611
Ain Shams University
Ain Shams Engineering Journal
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ENGINEERING PHYSICS AND MATHEMATICS
Numerical computation of nonlinear shock wave
equation of fractional order
* Corresponding author. Tel.: +91 9460905223.
E-mail addresses: [email protected] (D. Kumar),
[email protected] (J. Singh), [email protected]
(S. Kumar), [email protected] (Sushila), singhbhanu77@gmail.
com (B.P. Singh).
Peer review under responsibility of Ain Shams University.
Production and hosting by Elsevier
http://dx.doi.org/10.1016/j.asej.2014.10.0152090-4479 � 2014 Production and hosting by Elsevier B.V. on behalf of Ain Shams University.This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).
Devendra Kumar a,*, Jagdev Singh b, Sunil Kumar c, Sushila d, B.P. Singh e
a Department of Mathematics, JECRC University, Jaipur, 303905 Rajasthan, Indiab Department of Mathematics, Jagan Nath University, Jaipur, 303901 Rajasthan, Indiac Department of Mathematics, National Institute of Technology, Jamshedpur, 831014 Jharkhand, Indiad Department of Physics, Yagyavalkya Institute of Technology, Jaipur, 302022 Rajasthan, Indiae Department of Mathematics, IEC College of Engineering & Technology, Greater Noida, 201306 Uttar Pradesh, India
Received 5 July 2014; revised 26 September 2014; accepted 19 October 2014
Available online 8 December 2014
KEYWORDS
Laplace transform method;
Homotopy analysis method;
Homotopy analysis
transform method;
Maple code;
Fractional nonlinear shock
wave equation
Abstract The main aim of the present paper was to present a user friendly approach based on
homotopy analysis transform method to solve a time-fractional nonlinear shock wave equation
arising in the flow of gases. The proposed technique presents a procedure of constructing the set
of base functions and gives the high-order deformation equations in a simple form. The auxiliary
parameter ⁄ in the homotopy analysis transform method solutions has provided a convenient
way of controlling the convergence region of series solutions. The method is not limited to the small
parameter, such as in the classical perturbation method. The technique gives an analytical solution
in the form of a convergent series with easily computable components, requiring no linearization or
small perturbation. The numerical solutions obtained by the proposed approach indicate that the
approach is easy to implement and computationally very attractive.� 2014 Production and hosting by Elsevier B.V. on behalf of Ain Shams University. This is an open access
article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).
1. Introduction
The mathematical model of shock waves arose in connectionwith problems of the motion of gases and compressible fluids
around the second half of the 19th century. The pioneering
work in this area was conducted by Earnshaw [1], Riemann[2], Rankine [3] and Hugoniot [4]. The common features ofindividual theories of continuum physics include conservationlaws. The laws are supplemented by constitutive relations
which characterize the particular medium in question by relat-ing the values of the main vector field u to the flux, f. The lawsare supplemented by constitutive relations which characterize
the particular medium in question by relating the values ofthe main vector field u to the flux f.
Here we assume that these relations are represented by
smooth forms, and as a result the conservation laws lead tononlinear hyperbolic partial differential equations, which aregiven in simplest form as
606 D. Kumar et al.
utðx; tÞ þ fðuðx; tÞÞx ¼ 0; x 2 R; t > 0 ð1Þ
with the initial condition
uðx; 0Þ ¼ u0ðxÞ; x 2 R: ð2Þ
Eq. (1) occurs in a model for a diverse range of physical phe-
nomena from shock waves to three-phase flow in porousmedia. Shock waves find their applications in explosions, traf-fic flow, glacier waves, airplanes breaking the sound barrier
and so on. They are formulated by nonlinear hyperbolic par-tial differential equations [5–7]. The shock wave and waveequations have been studied using Adomian decomposition
method (ADM) [8] and homotopy perturbation method(HPM) [9]. The classical shock wave and wave equations arevery handy predictive equations, since solutions are easilyobtained. In this article, we present a fractional generalization
of shock wave Eq. (1), which can be written in the followingform
Dat uðx; tÞ þ fðuðx; tÞÞx ¼ 0; x 2 R; t > 0 ð3Þ
where a is a parameter describing the order of time-derivative.The derivative is considered in the Caputo sense. The generalresponse expression contains a parameter describing the order
of fractional shock wave equation that can be varied to obtainvarious responses. In the case of a = 1, the fractional shockwave equation reduces to the classical shock wave equation.
The most important advantage of using fractional differentialequations in mathematical modeling is their non-local prop-erty. It is well known that the integer order differential opera-
tor is a local operator but the fractional order differentialoperator is non-local. This means that the next state of a sys-tem depends not only upon its current state but also upon all
of its historical states. This is more realistic and it is one reasonwhy fractional calculus has become more and more popular[10–18]. There are several methods to solve the fractionalshock wave and wave equations, such as the HPM [19]. In
2014, Singh et al. [20] considered the fractional models ofshock wave and wave equations and proposed an analyticalalgorithm based on homotopy perturbation transform method
(HPTM). The homotopy analysis method (HAM) was firstproposed and developed by Liao [22–25] based on homotopy,a fundamental concept in topology and differential geometry.
The HAM has been successfully applied by many researchworkers to study the physical problems [26–31]. The Laplacetransform is a powerful technique for solving various linearpartial differential equations having considerable significance
in various fields such as engineering and applied sciences. Cou-pling of semi-analytical methods with Laplace transform giv-ing time consuming consequences and less C.P.U time
(Processor 2.65 GHz or more and RAM-1 GB or more) forsolving nonlinear problems. Recently, many researchers havepaid attention for solving the linear and nonlinear partial dif-
ferential equations using various methods combined with theLaplace transform. Among these are Laplace decompositionmethod (LDM) [32–34], homotopy perturbation transform
method [35–37] and homotopy analysis transform method[38–40].
In this article, we implement the HATM to find the analyt-ical and numerical solutions of the nonlinear fractional shock
wave and wave equations with time-fractional derivatives. Theproposed technique solves the nonlinear problems withoutusing Adomian polynomials and He’s polynomials which can
be considered as a clear advantage of this algorithm overdecomposition and the homotopy perturbation transformmethods. The plan of our paper is as follows: The basic defini-
tions of fractional calculus are given in Section 2. The basicidea of HATM is presented in Section 3. In Section 4, numer-ical examples is given to illustrate the applicability of the con-
sidered method. Section 5 is dedicated to numerical results anddiscussion. Conclusions are presented in Section 6.
2. Basic definitions of fractional calculus
In this section, we mention the following basic definitions offractional calculus.
Definition 1.1. The Laplace transform of the function f(t) is
defined by
FðsÞ ¼ L½ fðtÞ� ¼Z 1
0
e�stfðtÞdt: ð4Þ
Definition 1.2. The Laplace transform of the Riemann–Liou-ville fractional integral is defined as [12]:
L Iat uðx; tÞ� �
¼ s�aL½uðx; tÞ�: ð5Þ
Definition 1.3. The Laplace transform of the Caputo fractionalderivative is defined as [12]:
L Dat uðx;tÞ
� �¼ saL½uðx; tÞ��
Xn�1k¼0
sða�k�1ÞuðkÞðx;0Þ; n�1< a6 n:
ð6Þ
3. Analysis of the method
The HAM was first proposed by Liao [21] based on homotopy,a fundamental concept in topology. The HAM is based onconstruction of homotopy which continuously deform an ini-
tial guess approximation to the exact solution of the givenproblem. An auxiliary linear operator is chosen to constructthe homotopy and an auxiliary linear parameter is used to con-trol the region of convergence of the solution series. The
HATM is a combined form of the Laplace transform methodand the HAM. We apply the HATM to the following generalfractional nonlinear nonhomogeneous partial differential
equation of the form:
Dat uðx; tÞ þ Ruðx; tÞ þNuðx; tÞ ¼ gðx; tÞ; n� 1 < a 6 n ð7Þ
where Dat uðx; tÞ is the Caputo fractional derivative of the func-
tion uðx; tÞ;R is the linear differential operator, N representsthe general nonlinear differential operator and gðx; tÞ, is thesource term.
Applying the Laplace transform on both sides of Eq. (7),we get
L Dat u
� �þ L½Ru� þ L½Nu� ¼ L½gðx; tÞ�: ð8Þ
Using the differentiation property of the Laplace transform,we have
saL½u� �Xn�1k¼0
sa�k�1uðkÞðx; 0Þ þ L½Ru� þ L½Nu� ¼ L½gðx; tÞ�: ð9Þ
Numerical computation of nonlinear shock wave equation of fractional order 607
On simplifying
L½u�� 1
sa
Xn�1k¼0
sa�k�1uðkÞðx;0Þþ 1
sa½L½Ru�þL½Nu��L½gðx;tÞ�� ¼ 0:
ð10ÞWe define the nonlinear operator
N½/ðx; t;qÞ� ¼L½/ðx; t;qÞ�� 1
sa
Xn�1k¼0
sa�k�1/ðkÞðx;0;qÞþ 1
sa
�½L½R/ðx; t;qÞ�þL½N/ðx; t;qÞ��L½gðx; tÞ��; ð11Þwhere q 2 ½0; 1� and /ðx; t; qÞ are real functions of x, t and q.We construct a homotopy as follows
ð1� qÞL½/ðx; t; qÞ � u0ðx; tÞ� ¼ �hqHðx; tÞN½uðx; tÞ�; ð12Þ
where L denotes the Laplace transform, q 2 ½0; 1� is theembedding parameter, H(x, t) denotes a nonzero auxiliaryfunction, �h „ 0 is an auxiliary parameter, u0(x, t) is an initialguess of u(x, t) and /ðx; t; qÞ is an unknown function. Obvi-
ously, when the embedding parameter q= 0 or q= 1, it holds
/ðx; t; 0Þ ¼ u0ðx; tÞ; /ðx; t; 1Þ ¼ uðx; tÞ; ð13Þ
respectively. Thus, as q increases from 0 to 1, the solution/ðx; t; qÞ varies from the initial guess u0(x, t) to the solution
u(x, t). Expanding /ðx; t; qÞ in Taylor series with respect toq, we have
/ðx; t; qÞ ¼ u0ðx; tÞ þX1m¼1
umðx; tÞqm; ð14Þ
where
umðx; tÞ ¼1
m!
@m/ðx; t; qÞ@qm
jq¼0: ð15Þ
If the auxiliary linear operator, the initial guess, the auxiliary
parameter ⁄, and the auxiliary function are properly chosen,the series (14) converges at q = 1, then we have
uðx; tÞ ¼ u0ðx; tÞ þX1m¼1
umðx; tÞ; ð16Þ
which must be one of the solutions of the original nonlinearequations. According to the definition (16), the governingequation can be deduced from the zero-order deformation
(12).Define the vectors
~um ¼ fu0ðx; tÞ; u1ðx; tÞ; . . . ; umðx; tÞg: ð17Þ
Differentiating the zeroth-order deformation Eq. (12) m-times
with respect to q and then dividing them by m! and finally set-ting q = 0, we get the following mth-order deformationequation:
L½umðx; tÞ � vmum�1ðx; tÞ� ¼ �hqHðx; tÞRmð~um�1Þ: ð18Þ
Applying the inverse Laplace transform, we have
umðx; tÞ ¼ vmum�1ðx; tÞ þ �hL�1½qHðx; tÞRmð~um�1Þ�; ð19Þ
where
Rmð~um�1Þ ¼1
ðm� 1Þ!@m�1N½/ðx; t; qÞ�
@qm�1jq¼0; ð20Þ
and
vm ¼0; m 6 1;
1; m > 1:
�ð21Þ
4. Numerical examples
In this section, we discuss the implementation of our numericalmethod and investigate its accuracy and stability by applying it
to numerical examples on time-fractional shock wave andwave equations.
Example 1. Consider the following time-fractional shock waveequation
Dat u�
1
c0� cþ 1
2
u
c20
� �Dxu ¼ 0; ðx; tÞ
2 R� ½0;T�; 0 < a 6 1; ð22Þ
where c0; c are constants and c is the specific heat. For studyingthe case under consideration, we take c0 = 2 and
c = 1.5, which corresponds to the flow of air, subject to theinitial condition
uðx; 0Þ ¼ exp � x2
2
� �: ð23Þ
Applying the Laplace transform on both sides of Eq. (22) sub-
ject to the initial condition (23), we have
L½uðx; tÞ��1
sexp �x
2
2
� �� 1
saL
1
c0Dxu�
cþ1
2
u
c20Dxu
� �¼ 0: ð24Þ
We define a nonlinear operator as
N½/ðx;t;qÞ�¼L½/ðx;t;qÞ��1sexp �x
2
2
� �
� 1
saL
1
c0Dx/ðx;t;qÞ�
cþ12c20
/ðx;t;qÞDx/ðx;t;qÞ� �
¼0: ð25Þ
and thus
Rmð~um�1Þ ¼ L½um�1� � ð1� vmÞ1
sexp � x2
2
� �
� 1
saL
1
c0Dxum�1 �
cþ 1
2c20
Xm�1r¼0
urDxum�1�r
" #: ð26Þ
The mth-order deformation equation is given by
L½umðx; tÞ � vmum�1ðx; tÞ� ¼ �hRmð~um�1Þ: ð27Þ
Applying the inverse Laplace transform, we have
umðx; tÞ ¼ vmum�1ðx; tÞ þ �hL�1½Rmð~um�1Þ�: ð28ÞWe start with the initial condition u0ðx; tÞ ¼ uðx; 0Þ ¼exp � x2
2
, use the iterative scheme (28) and obtain the various
iterates
u1ðx; tÞ¼ �h1
c0� cþ1
2c20exp �x
2
2
� �� �x exp �x
2
2
� �ta
Cðaþ1Þ ; ð29Þ
u2ðx; tÞ¼ �hð1þ�hÞ 1
c0� cþ1
2c20exp �x
2
2
� �� �xexp �x
2
2
� �ta
Cðaþ1Þþ�h2 exp �x2
2
� �� 1
c20þx2
c20þðcþ1Þ
c30exp �x
2
2
� ��
�2ðcþ1Þc30
x2 exp �x2
2
� ��ðcþ1Þ2
4c40expð�x2Þ
þ3ðcþ1Þ2
4c40x2 expð�x2Þ
#t2a
Cð2aþ1Þ : ð30Þ
Figure 1 Plot of u(x, t) w.r. to x and t at a = 1/4 for Example 1.
608 D. Kumar et al.
Proceeding in the same manner the rest of components of the
HATM solution can be obtained. Finally, we have the seriessolution
uðx; tÞ ¼ u0ðx; tÞ þ u1ðx; tÞ þ u2ðx; tÞ þ � � � : ð31Þ
However, most of the results given by theHPMandHPTMcon-verge to the corresponding numerical solutions in a rather smallregion. But, different from those two methods, the HATM pro-
vides us with a simple way to adjust and control the convergenceregion of solution series by choosing a proper value for the aux-iliary parameter ⁄. Choosing ⁄ = �1 in (31) we can recover all
the results obtained by HPM [19] and HPTM [20].
Example 2. Next, consider the following time-fractional waveequation
Dat uþ uDxu�Dxxtu ¼ 0; ð32Þ
with the initial condition
uðx; 0Þ ¼ 3 sec h2x� 15
2
� �: ð33Þ
Operating the Laplace transform on both sides of Eq. (32) sub-ject to the initial condition (33), we have
L½uðx; tÞ� � 3
ssec h2
x� 15
2
� �þ 1
saL½uDxu�Dxxtu� ¼ 0: ð34Þ
We define a nonlinear operator as
N½/ðx; t; qÞ� ¼ L½/ðx; t; qÞ� � 3
ssec h2
x� 15
2
� �
þ 1
saL½/ðx; t; qÞDx/ðx; t; qÞ �Dxxt/ðx; t; qÞ� ¼ 0: ð35Þ
and thus
Rmð~um�1Þ ¼ L½um�1� � ð1� vmÞ3
ssec h2
x� 15
2
� �
þ 1
saLXm�1r¼0
urDxum�1�r �Dxxtum�1
" #: ð36Þ
The mth-order deformation equation is given by
L½umðx; tÞ � vmum�1ðx; tÞ� ¼ �hRmð~um�1Þ: ð37Þ
Applying the inverse Laplace transform, we have
umðx; tÞ ¼ vmum�1ðx; tÞ þ �hL�1½Rmð~um�1Þ�: ð38Þ
Using the initial approximation u0ðx; tÞ ¼ uðx; 0Þ ¼3 sec h2 x�15
2
� �and the iterative scheme (38), obtain the various
components of series solution
u1ðx; tÞ ¼ �9�h sec h4x� 15
2
� �tanh
x� 15
2
� �ta
Cðaþ 1Þ ; ð39Þ
u2ðx; tÞ ¼ �9�hð1þ �hÞ sec h4x� 15
2
� �tanh
x� 15
2
� �ta
Cðaþ 1Þ
þ �h2189
2sec h6
x� 15
2
� �tanh2 x� 15
2
� ���
� 27
2sec h6
x� 15
2
� ��t2a
Cð2aþ 1Þ
þ 135
2sec h4
x� 15
2
� �tanh3 x� 15
2
� ��
� 63
2sec h
4 x� 15
2
� �tanh
x� 15
2
� ��Cðaþ 1Þt2a�1
Cð2aÞ
: ð40Þ
In the similar way, the rest of components of the HATM solu-tion can be obtained. Thus the series solution u(x, t) of the Eq.(32) is given as:
uðx; tÞ ¼ u0ðx; tÞ þ u1ðx; tÞ þ u2ðx; tÞ þ � � � : ð41Þ
If we set ⁄ = �1 and a = 1 in (41), then the obtained solutionconverges to the exact solution
uðx; tÞ ¼ 3 sec h2x� 15� t
2
� �: ð42Þ
We can also recover all the results obtained by HPM [19] andHPTM [20], just by choosing ⁄ = �1 in (41).
5. Numerical results and discussion
In this section, we calculate the numerical solutions of theprobability density function u(x, t) for different time-fractional
Brownian motions a ¼ 14; 12and also the standard motion
a = 1.The numerical values u(x, t) vs. time t and also those vs. x
at a = 1 for both the two examples are calculated for different
particular cases and are depicted through Figs. 1–8. Duringnumerical computation only two iterations are considered. Itis evident that by using more terms, the accuracy of the results
can be dramatically improved and the errors converge to zero.The time-fractional shock wave equation considered in
Example 1 is graphically represented through Figs. 1–4. The
numerical results of the probability density function u(x, t)for fractional Brownian motion a ¼ 1
4with c = 1.5 and
c0 = 2, is shown through Fig. 1 and those for different valuesof t and a at x= 1 are depicted in Fig. 2. The numerical values
u(x, t) vs. time t and also those vs. x at a = 1, with c = 1.5 andc0 = 2, are depicted through Figs. 3 and 4. It is observed fromFig. 2 that as the value of a increase, the value of u(x, t)
increases but afterward its nature is opposite. It is also seenfrom Fig. 3 that u(x, t) increases with the decrease in t. Wecan also observe from Fig. 4 that u(x, t) is highest at x = 0
and decreases as numeric value of x increases.The wave equation with time-fractional derivative consid-
ered in Example 2 is described through Figs. 5–8. It is observed
from Fig. 5 that u(x, t) increases with the increase in x anddecreases with the increase in t for a = 1/4.
Figure 2 Plots of u(x, t) vs. t at x= 1 for different values of a for
Example 1.
Figure 4 Plot of u(x, t) vs. x at t= 1 and a = 1 for Example 1. Figure 7 Plot of u(x, t) vs. x at t= 1 and a = 1 for Example 2.
Figure 5 Plot of u(x, t) w.r. to x and t at a = 1/4 for Example 2.
Figure 3 Plot of u(x, t) vs. t at x = 1 and a = 1 for Example 1. Figure 6 Plots of u(x, t) vs. t at x = 1 for different values of a for
Example 2.
Numerical computation of nonlinear shock wave equation of fractional order 609
Figure 8 Plot of u(x, t) vs. t at x = 4 and a = 1 for Example 2.
610 D. Kumar et al.
It is observed from Fig. 6 that as the value of a increase, the
value of u(x, t) increases but afterward its nature is opposite. Itcan be seen from the Figs. 7 and 8 that u(x, t) increases withthe increase in x and decreases with the increase in t.
6. Concluding remarks
In this paper, our main concern has been to study the time-fractional shock wave and wave equations arising in flow of
gases. The HATM and symbolic calculations have been usedto obtain the approximate analytic solutions of the time-frac-tional shock wave and wave equations. The results obtained
by using the scheme presented here agree well with the analyt-ical solutions and the numerical results obtained by HPM [19]and HPTM [20]. It provides us with a simple way to adjust and
control the convergence region of solution series by choosingproper values for auxiliary parameter �h. The proposed methodprovides the solutions in terms of convergent series with easily
computable components in a direct way without using lineari-zation, perturbation or restrictive assumptions. Thus, it can beconcluded that the HATM is powerful and efficient in findinganalytical as well as numerical solutions for wide classes of
nonlinear fractional differential equations arising in science,engineering and finance.
Acknowledgment
The authors are extending their heartfelt thanks to the review-
ers for their valuable suggestions for the improvement of thearticle.
References
[1] Earnshaw S. On the mathematical theory of sound. Phil Trans
Roy Soc Lond 1860;150:133–48.
[2] Riemann B. Ueber die Fortpflanzung ebener Luftwellen von
endlicher Schwingungsweite, Gesamm. Math. Werke, Dover,
Reprint; 1953. p. 156–75.
[3] Rankine WJM. On the thermodynamic theory of waves of finite
longitudinal disturbance. Phil Trans Roy Soc Lond 1870;160:
277–88.
[4] Hugoniot H. Memoire sur la propagation du mouvement dans les
corps et plus specialement dans les gaz parfaits, 2e Partie. J Ecole
Polytech (Paris) 1889;58:1–125.
[5] Behrens J. Atmospheric and ocean modeling with an adaptive
finite element solver for the shallow-water equations. Appl Numer
Math 1998;26(1–2):217–26.
[6] Bermudez A, Vazquez ME. Upwind methods for hyperbolic
conservation laws with course terms. Comput Fluids 1994;23:
1049–71.
[7] Bruno R. Convergence of approximate solutions of the Cauchy
problem for a 2 · 2 non-strictly hyperbolic system of conservation
laws. In: Shearer M, editor. Nonlinear hyperbolic problems.
Taormina: Vieweg Braunschweig; 1993. p. 487–94.
[8] Allan FM, Al-Khaled K. An approximation of the analytic
solution of the shock wave equation. J Comput Appl Math
2006;192:301–9.
[9] Berberler ME, Yildirim A. He’s homotopy perturbation method
for solving the shock wave equation. Appl Anal 2009;88(7):
997–1004.
[10] Young GO. Definition of physical consistent damping laws with
fractional derivatives. Z Angew Math Mech 1995;75:623–35.
[11] Hilfer R, editor. Applications of fractional calculus in physics.
Singapore-New Jersey-Hong Kong: World Scientific Publishing
Company; 2000. p. 87–130.
[12] Podlubny I. Fractional differential equations. New York: Aca-
demic Press; 1999.
[13] Mainardi F, Luchko Y, Pagnini G. The fundamental solution of
the space-time fractional diffusion equation. Fract Calc Appl
Anal 2001;4:153–92.
[14] Debnath L. Fractional integrals and fractional differential equa-
tions in fluid mechanics. Frac Calc Appl Anal 2003;6:119–55.
[15] Caputo M. Elasticita e Dissipazione. Bologna: Zani-Chelli; 1969.
[16] Miller KS, Ross B. An introduction to the fractional calculus and
fractional differential equations. New York: Wiley; 1993.
[17] Oldham KB, Spanier J. The fractional calculus theory and
applications of differentiation and integration to arbitrary order.
New York: Academic Press; 1974.
[18] Kilbas AA, Srivastava HM, Trujillo JJ. Theory and applications
of fractional differential equations. Amsterdam: Elsevier; 2006.
[19] Singh M, Gupta PK. Homotopy perturbation method for time-
fractional shock wave equation. Adv Appl Math Mech 2011;3(6):
774–83.
[20] Singh J, Kumar D, Kumar S. A new fractional model of nonlinear
shock wave equation arising in flow of gases. Nonlinear Eng
2014;3(1):43–50.
[21] Liao SJ. Beyond perturbation: introduction to homotopy analysis
method. Boca Raton: Chapman and Hall/CRC Press; 2003.
[22] Liao SJ. On the homotopy analysis method for nonlinear
problems. Appl Math Comput 2004;147:499–513.
[23] Liao SJ. A new branch of solutions of boundary-layer flows over
an impermeable stretched plate. Int J Heat Mass Trans
2005;48:2529–39.
[24] Liao SJ. An approximate solution technique not depending on
small parameters: a special example. Int J Non-linear Mech
1995;30(3):371–80.
[25] Liao SJ. Comparison between the homotopy analysis method and
homotopy perturbation method. Appl Math Comput 2005;169:
1186–94.
[26] Hayat T, Khan M, Ayub M. On the explicit solutions of an
Oldroyd 6-constant fluid. Int J Eng Sci 2004;42:125–35.
[27] Hayat T, Khan M. Homotopy solution for generalized second-
grade fluid fast porous plate. Nonlinear Dynam 2005;4(2):
395–405.
[28] Shidfar A, Molabahrami A. A weighted algorithm based on the
homotopy analysis method: application to inverse heat conduc-
tion problems. Commun Nonlinear Sci Numer Simul 2010;15:
2908–15.
Numerical computation of nonlinear shock wave equation of fractional order 611
[29] Abbasbandy S, Shivanian E, Vajravelu K. Mathematical proper-
ties of h-curve in the frame work of the homotopy analysis
method. Commun Nonlinear Sci Numer Simul 2011;16:4268–75.
[30] Abbasbandy S. Homotopy analysis method for the Kawahara
equation. Nonlinear Anal Real World Appl 2010;11:307–12.
[31] Abbasbandy S. Approximate solution for the nonlinear model of
diffusion and reaction in Porous catalysts by means of the
homotopy analysis method. Chem Eng J 2008;136:144–50.
[32] Khuri SA. A Laplace decomposition algorithm applied to a class
of nonlinear differential equations. J Appl Math 2001;1:141–55.
[33] Khan M, Hussain M. Application of Laplace decomposition
method on semi-infinite domain. Numer Algorithms 2011;56:
211–8.
[34] Khan M, Gondal MA, Kumar S. A new analytical solution
procedure for nonlinear integral equations. Math Comput Model
2012;55:1892–7.
[35] Gondal MA, Khan M. Homotopy perturbation method for
nonlinear exponential boundary layer equation using Laplace
transformation He’s polynomials and Pade technology. Int J
Nonlinear Sci Numer Simul 2010;11:1145–53.
[36] Kumar S. A numerical study for solution of time fractional
nonlinear shallow-water equation in oceans. Zeitschrift fur
Naturforschung A 2013;68a:1–7.
[37] Singh J, Kumar D, Kumar S. New treatment of fractional
Fornberg–Whitham equation via Laplace transform. Ain Sham
Eng J 2013;4:557–62.
[38] Khan M, Gonda MA, Hussain I, Karimi Vanani S. A new
comparative study between homotopy analysis transform method
and homotopy perturbation transform method on semi-infinite
domain. Math Comput Model 2012;55:1143–50.
[39] Kumar D, Singh J, Sushila. Application of homotopy analysis
transform method to fractional biological population model.
Roman Reports Phys 2013;65(1):63–75.
[40] Salah A, Khan M, Gondal MA. A novel solution procedure for
fuzzy fractional heat equations by homotopy analysis transform
method. Neural Comput Appl 2013;23(2):269–71.
Devendra Kumar is an Assistant Professor-I in
the Department of Mathematics, JECRC
University, Jaipur-303905, Rajasthan, India.
His area of interest is special functions, frac-
tional calculus, differential equations, integral
equations, homotopy methods, Laplace
transform method and Sumudu transform
method.
Jagdev Singh is an Assistant Professor in the
Department of Mathematics, Jagan Nath
University, Jaipur-303901, Rajasthan, India.
His fields of research include special functions,
fractional calculus, fluid dynamics, homotopy
methods, Adomian decomposition method,
Laplace transform method and Sumudu
transform method.
Sunil Kumar is an Assistant Professor in the
Department of Mathematics, National Insti-
tute of Technology, Jamshedpur, 801014,
Jharkhand, India. His fields of research
include singular integral equation, fractional
calculus, homotopy methods, Adomian
decomposition method and Laplace transform
method.
Sushila is an Assistant Professor in the
Department of Physics, Yagyavalkya Institute
of Technology, Jaipur-302022, Rajasthan,
India. Her fields of research include nuclear
physics, solid state physics, plasma physics,
computational fluid dynamics, fractional cal-
culus, homotopy methods, Laplace transform
method and Sumudu transform method.
B.P. Singh is an Assistant Professor in the
Department of Mathematics, IEC College of
Engineering & Technology, Greater Noida,
201306, Uttar Pradesh, India. His fields of
research include singular integral equation,
fractional calculus, homotopy methods, Ado-
mian decomposition method and Laplace
transform method.